We report the development of a stable high-average power X-ray source generated by the interaction of ultrashort laser pulses (35 fs, 1 mJ, 1 kHz) with a solid target in air. The achieved source stability, which is essential for the applications foreseen for these laser-driven plasma accelerators, is due to the combination of precise positioning of the target on focus and the development of a fast rotating target system able to ensure the refreshment of the material at every shot while minimizing positioning errors with respect to the focal spot. This vacuum-free laser-plasma X-ray source provides an average dose rate of 1.5 Sv/h at 30 cm and a repeatability better than 93% during more than 36 min of continuous operation per target.